One of the most important results concerning gcd is the following. This is so important that sometimes it is taken as the definition of the gcd.
Result: If g is the gcd of a and b then there exists integers X and Y such that g = Xa + Yb. Also g is the least positive number expressible in this form xa + yb.
Proof: Consider the set S = {xa+yb: x,y are integers}. Choose d to be the least positive integer of this set, so that d = Xa + Yb for some integers X and Y. We shall now show that d = g. For this we shall first show that d divides both a and b. We'll in fact show only that d divides a. The other case is similar.
To prove that d divides a assume the contrary is true, i.e. d does not divide a. Then application of the division algorithm for d and a gives a non zero remainder, i.e. a = dq + r where 0 < r < d. Hence we have r = a - dq = a - q(Xa + Yb) = (1-qX)a + (-qY)b. So r is a positive number in S by virtue of having the desired form of elements in S. But then 0 < r < d contradicts the fact that d was chosen to be the least positive integer of S. So r must be 0 and so d must divide a.
Next we show that g = d. Note that d is a common divisor of a and b and so d ≤ g by definition of g. But as g divides both a and b it also divides their linear combinations so that g divides Xa + Yb i.e. g divides d. Both integers being positive this implies that g ≤ d. Hence g = d.
By virtue of this result you are entitled to think of the gcd as the least positive number among all numbers having the form xa + yb. Clearly if any combination of a and b is 1 that is the least positive possible and so obviously their gcd is 1. This is what happens in the case of a and a-1.
Write back if there are further doubts.